SOALAN RAMALAN 2011
STATISTICS
1. Table 14 shows marks obtained by a group of students in a monthly test.
Marks | 1 – 20 | 21 - 40 | 41 – 60 | 61 – 80 |
Number of students | 6 | 9 | 13 | 12 |
Table 14
Without drawing an ogive, find the third quartile mark.
2. The mean of 5 numbers is 2∫h. The sum of the square of the numbers is 605 and the variance is 3k2. Express h in terms of k.
3. Table 22 shows the marks of a group of students in an examination.
Marks | 1 | 2 | 3 | 4 | 5 |
Number of students | 4 | 6 | 8 | x | 7 |
Find
(a) The maximum value of x if the mode mark is 8.
(b) The minimum value of x if the mean of the marks is greater than 3
4. The mean of six numbers is ∫2m. The sum of the squares of the numbers is 120 and the standard deviation is 4k.
5. The mean of four numbers is ∫3k. The sum of the squares of the numbers is 112 and the variance is m.
6. Set X consists of 50 scores, x for a game which has a mean of 6 and standard deviation of 4.5.
Calculate
(a) The sum of the scores ∑ x
(b) The sum of the squares of the scores ∑x2
7. Table 22 shows the scores of a group of students in a quiz; calculate the standard deviation of the scores.
Score | 1 | 2 | 3 | 4 |
Number of students | 4 | 23 | 10 | 5 |
Table 22
8. A set of data which consists of 12 numbers has a mean of m and variance of 4. The sum of the numbers is 360. Find
(a) The value of m
(b) The sum of squares of the numbers
9. Given that the mean and variance of a set of n numbers x1, x2,…….., xn are 3 and 2.56 respectively. Find the mean and standard deviation of the new set of n numbers 5x1 – 2, 5x2 – 2, …….., 5xn – 2.
10. The mean of six numbers is ∫2m. The sum of the squares of the numbers is 120 and the standard deviation is 4k. Express m in terms of k.
11. A set of data consists of 6 numbers. The sum of the numbers is 76 and the sum of squares of the numbers is 1064, find
(a) The mean
(b) The standard deviation
12. A set of 5 numbers has a mean of 8.
(a) Find ∑x
(b) When a number m is added to this set, the new mean is 9.5. Find the value of m.
13. A set of eight scores x1, x2, x3, …….., x8 has mean 7 and standard deviation 2. Find
(a) ∑x
(b) ∑x2
14. A set of quiz score x1, x2, x3, x4, x5, x6 has mean 6 and standard deviation 2. If every score is multiply by 3 and then minus 2, find from the new set of the score.
(a) The mean
(b) The variance
15. A set of data consists of four numbers. The sum of the numbers is 40 and the sum of the squares of the numbers is 454. Find, for the four numbers.
(a) The mean
(b) The standard deviation
16. The mean of 5 numbers is ∫20. The sum of the squares of the numbers is 25k and the standard deviation is p. Express k in the terms of p.
17. A set of six numbers has a mean of 177.
(a) Find ∑x
(b) When a number q is added to this set, the new mean is 180. Find the value of q.
18. A set of positive integers consists of 2, 3, p and 6. The variance for this set of integers is 2.5, find the value of p.
19. A set of numbers x1, x2, x3, x4, ………, xn has a median of 5 and a standard deviation of 2. Find the median and the variance for the set numbers 6x1 + 1, 6x2 + 1, 6x3 + 1, …….., 6xn + 1.
20.
The standard deviation of a set of numbers m, 2m, 3m, 4m and 5m is p. Find the mean x in terms of m and p.
21. A set of data consists of six numbers. The sum of the numbers is 70 and the sum of the squares of the numbers is 960. A number m is added to the data. The mean of seven numbers is 12. Find
(a) The value of m
(b) The standard deviation for the seven numbers.
22. A set of data consists of five numbers. The sum of the numbers is 30 and the sum of the squares of the numbers is 225.
(a) Find the mean for the five numbers.
(b) When a number p is added to this set, the mean is unchanged. Hence, find the variance.
23. Table 21 shows the frequency distribution of ages of workers.
Age(Years) | 28 – 32 | 33 – 37 | 38 – 42 | 43 – 47 | 48 – 52 |
Numbers of workers | 16 | 38 | 26 | 11 | 9 |
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Table 21
Given the third quartile of ages of workers is K = L + ( )5 find the values of K, L, G and F.
24. A set of data: 65, 40, 65, 50, p, 82, 73 and 50 has a mean of 60. Find
(a) The value of p
(b) The standard deviation of the set of data
25. A set of positive integers consists of 2, 5 and k. the standard deviation for this set of integers is given as ∫6. Find the value of k.
26. Given a set of 5 numbers x1, x2, x3, x4,and x5, has a mean of 20 and the standard deviation of 2.5.
For the following set of data 3x1 + q, 3x2 + q, 3x3 + q, 3x4 + q, and 3x5 + q, find
(a) The value of q, if the mean is 70
(b) The variance
27. A set of five numbers has a mean of 8 and a standard deviation of ∫2. Find
(a) ∑x
(b) ∑x2
28. The set of data 6, 8, 14, 6, 12, 14, 5, p, q has a mean of 9, mode of 6 and it is given p > q. Find
(a) The values of p and q
(b) The median
1. From the quadratic equation which has the roots -3 and ¼. Give the answers in the form ax2 + bx + c = 0 where a, b and c are constants.
2. Given that 3 and k are the roots of the quadratic equation x2 + x = p. Find the value of k and p.
3. Solve the quadratic equation (3x – 5)x = 2x + 1. Give your answers correct to three decimal places.
4. Given that the roots of a quadratic equation are 2/3 and -4. Form the quadratic equation and write it in the form ax2 + bx + c = 0 where a, b and c are constants.
5. Write the quadratic equation 2x2 – 4x = 3x2 + 7x – 15 in general form. Then, solve it by using formula. Give your answer correct to 3 decimal places.
6. If one of the roots of the quadratic equation 2x2 – px = 12 is 3
a. The value of p
b. The other root
7. Find the value of h if the quadratic equation hx2 – 4 + 3 = 0 has two equal roots.
8. The quadratic equation x2 – (3 – p)x + p – 3 = 0, where p is constant, has two equal roots. Find the possible values of p.
9.
(a) Express the quadratic equation 2(x + 1)2 = 5x + 3 in the general form.
(b) Given that 4 is one of the roots of the quadratic equation 2x2 – hx + 4 = 0, find the value of h
10. Given a and b are the roots of the quadratic equation 3x2 + 4x – 6 = 0, form the quadratic equation whose roots 3a and 3b.
11. Given that x = 2 and x = -1/3 are the roots of the equation 3x2 + bx + c = 0, find the value of b and the value of c.
12. Given that the straight line y = 4x + 1 is a tangent to the curve y = x2 + k. Find the value of k.
13. Find the value of h if the quadratic equation hx2 – 4x + 3 = 0 has two equal roots.
14. Form the quadratic equation which has the roots -3 and ¼ . Given the answer in the form ax2 + bx + c = 0, where a, b and c are constants.
15. Solve the quadratic equation (3x – 5)x = 2x + 1. Give your answers correct to three decimal places.
16. Write the equation (3x + 1)(x – 1) = x(x + 2) in general form.
17. If one of the roots of the quadratic equation x2 – kx – 10 = 0 is 2, calculate the value of k.
18. Form the quadratic equation which has roots -4 and 2/3. Give your answer in the form ax2 + bx + c = 0, where a, b, and c are constants.
19. The quadratic equation x2 – px + q = 0, where p and q are constants, has two equal roots. Express q in terms of p.
20. One of the roots of the equation 9x2 – 3x – k = 0 is two times the other root, find the possible value of k.
21. Given that 1 is one of the roots the quadratic equation
p px2 + 7x + 2p = 0, find the value of p.
22. If the straight line y = 1 – mx touches the curve y = x2- 4x +m at a point, determine the values of m.
23. The straight line y = p(1 – 2x) is a tangent to the curve y – x2 = 2. Find the possible value of p.
24. Solve the quadratic equation 2x2 – 6x = x(x + 3) – 4. Give your answer correct to four significant figures.
25. Solve the quadratic equation 2x (x – 4) = (1 – x)(x + 2). Give your answer correct to four significant figures.
26. Form a quadratic equation which has the roots 6 and -½ . Give your answer in the general form ax2 + bx + c = 0, where a, b and c are constants.
27. Given one of the roots of the quadratic equation x2 – px – 27 = 0 is the square of the other root.
a. Find the value of p
b. State the roots of the quadratic equation
28. Solve the quadratic equation 2x(x + 2) = 5(x + 1) + 2. Give your answer correct to four significant figures.
29. The straight line y = 3x + 2 touches the curve y = 2x2 – x + q. Find the value of q.
30. Solve the quadratic equation 2x(x – 3) = 3x + 1. Give your answer correct to three decimal places.
31. One of the roots of the equation x2 + px + 18 = 0 is half the other root. Find the possible values of p.
32. Solve the quadratic equation 3x + 1 = 2x(x – 3). Give your answer correct to four significant figures.
33. Given that m and 3m are the roots of the quadratic equation 2x2 + 4x + n = 0. Find the value of m and of n.
34. Solve the quadratic equation 3 – 8(x – 1) = 2x(x + 1). Give your answers correct to four significant figures.
35. Given that -5 is one of the roots of the quadratic equation 15 – px = 2x2. Find the value of p.
36. A quadratic equation (k + 3)x2 = 12x + 2k = 0 has two equal roots. Find the possible values of k.
Find the range of values of p if the quadratic equation
18x2 + 12x + 7 = p has two different roots.
Question 2
Solve the quadratic inequality x2 – 2x < 3
Question 3
Diagram 6 shows a quadratic function graph f (x) = h – 2(x + k)2 where h and k are constants.
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Diagram
(a) The value of h
(b) The value of k
(c) The value equation when the curve is reflected through x-axis
Question 4
Diagram 5 shows the graph of a quadratic function
f (x) = 3(x + p)2 + 5, where p is constants.
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The curve y = f(x) has the minimum point (-2 , q ) where q is a constant.
(a) The value of p
(b) The value of q
(c) The equation of the axis of symmetry
Question 5
Find the range of values of x for which (x – 3)(2x + 1) > x2 – 9
Question 6
Find the range of values of x for which (x – 3)2 ≤ 3
Question 7
Diagram 5 shows the graph of a quadratic function y = f (x).
The straight line y = 3 is a tangent to the curve y = f (x)
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(a) Write the equation of the axis of symmetry of the curve
(b) Express f(x) in the form (x + p)2 + q , where p and q are constants.
Question 8
The graph of the quadratic function y = -2(x – 3)2 + 11 has a maximum point (p , q) and y-intercept is h.
Find the value of
(a) p
(b) q
(c) h
Question 9
Find the range of values of x for which 6x – x(2 – 5x) ≤ 12
Question 10
Diagram 5 shows the graph of a quadratic function
f(x) = 3(x + p)2 + 2, where p is a constant. The curve y = f(x) has the minimum point (4, q), where q is a constants.
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State,
(a) The value of p
(b) The value of q
(c) The equation of the axis of symmetry
Question 11
Find the range of values of x for which x(x – 6) ≤ 27
Question 12
Find the range of values of x for which 4x2 + 3 < 7x
Question 13
Diagram 6 shows the graph of f(x) = a(x + p)2 + q, where a, p and q are constant. The curve y = f(x) has the maximum point (4, -5).
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State,
(a) The range of the value of a
(b) The value of p
(c) The equation of the axis of symmetry
Question 14
Find the range of values of k for which function f(x) = x2 + kx + 2k – 3 does not intersect the x-axis
Question 15
The quadratic function f(x) = x2 + 4x – 1 can be expressed in the form f(x) = (x – p)2 + q , where p and q are constants.
Find the value of p and q
Question 16
Find the range of values of x for which 5x > 2x2 – 3
Question 17
The quadratic function x(x + y) + 8 = 0 does not intersects the straight line x + 2y = p , where p is a constants.
Find the range of value of p
Question 18
Diagram 5 shows the graph of the function y = -(x – p)2 + 25/4, where p is a constant.
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Find
(a) The value of p
(b) The equation of the axis of symmetry
Question 19
Given a quadratic function f (x) = 8x + 2x2 = 2(x + hk)2 + k , where h and k are constants.
State the value of h and value of k
Question 20
Find the range of the values of n for which (1 + n)(6 – n) < -8
Question 21
Given that the graph of quadratic function f (x) = 2x2 + bx + 8 always lies above the x-axis. Find the range of values of b.
Index Number
Soalan Ramalan SPM 2011
-CIKGU VEN-
1. Table 3 shows the prices for four ingredients A, B C and D used in baking a particular kind of cake. Diagram 13 is a bar chart which represents the relative amount of the ingredients A, B, C and D used in baking these cakes.
| Ingredients | Price per kg (RM) | Price index for the year 2007 based on the year 2005 | |
| Year 2005 | Year 2007 | ||
| A | x | 8.00 | 160 |
| B | 2.00 | 2.50 | 125 |
| C | 6.00 | y | 115 |
| D | 7.00 | 8.40 | z |
Table 13
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Diagram 13
(a) Find the value of x, y and z.
(b) (i) Calculate the composite index for the cost of baking these cakes in the year 2007 based on the year 2005.
(ii) Hence, calculate the corresponding cost of baking these cakes in the year 2005 if the cost in year 2007 was RM 4600.
(c) The cost of baking these cakes is expected to increase by 40% from the year 2005 to the year 2008. Find the expected composite index for the year 2008 based on the year 2007.
2. Table 13 shows the prices, the price indices and percentage of usage of four items A, B, C and D which are the main ingredients in the manufacturing of a types of biscuits.
| Item | Price per unit (RM) | Price index for the year 2007 based on the year 2005 | Percentage of usage (%) | |
| 2005 | 2007 | |||
| A | p | 45 | 125 | 7m |
| B | 55 | q | 120 | 8m |
| C | 40 | 42 | 105 | 28 |
| D | 50 | 47 | r | 9m |
Table 13
(a) Find the value of p , of q and of r
(b) State the value of m. Hence, calculate the composite index for the cost of manufacturing the biscuits in the year 2007 based on the year 2005.
(c) Calculate the price of a box of biscuits in the year 2005 if the corresponding price in the year 2007 is RM 25.70
(d) The coast of manufacturing the biscuits is expected to increase by 20% from the year 2007 to the year 2009. Find the expected composite index for the year 2009 based on the year 2005.
3. Table 13 shows the price indices in the year 2009 based on the year 2007 of four items A, B, C and D used in the production of a cake.
| Item | Price index in the year 2009 based on the year 2007 | Weightage |
| A | 130 | 1 |
| B | 140 | 3 |
| C | 115 | 2n |
| D | 120 | n |
Table 13
(a) Given the price of A is RM 2.50 in the year 2009, calculate its price in 2007
(b) Given that the composite index for the year 2009 based on the year 2007 is 125. Find the value of n.
(c) Find the price of the cake in the year 2007 if its corresponding price in the year 2009 is RM 46.00
(d) Given that the price of item D is estimated to increase by 10% from the year 2009 to 2010 while the other items remain unchanged.
Calculate the composite index of the cake for the year 2010 based on the year 2007
4. A cake was made by using 4 special ingredients A, B, C and D. Table 3 shows the prices in the year 2007 and 2008, the price indices for the year 2008 based on the year 2007 and the number of parts of the ingredients used in making the cake.
| Ingredients | Price per unit (RM) | Price index | Number of parts | |
| 2007 | 2008 | |||
| A | 2.00 | 2.50 | x | 2 |
| B | y | 7.00 | 140 | 3 |
| C | 3.60 | 4.50 | 125 | 4 |
| D | 4.00 | z | 130 | 1 |
(a) Find the value of x, y and z
(b) Calculate
(i) The composite index for the production cost of a cake in the year 2008 based on the year 2007
(ii) The price of the cake in the year 2008 if its price in the year 2007 is RM 35
(c) The cost of all the ingredients increased by 20% from the year 2008 to the year 2009.
Calculate the composite index for the 2009 based on the year 2007.
5. Table 3 shows the price and price indices for four items P, Q, R and S. Diagram 7 shows a bar chart of the sales for these items for the year 2005.
| Item | Price (RM) | Price index for the year 2007 based on the year 2005 | |
| 2005 | 2007 | ||
| P | 1.25 | x | 120 |
| Q | 3.20 | 4.40 | y |
| R | 2.60 | 3.25 | 125 |
| S | z | 2.70 | 135 |
Table 3
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Diagram 7
(a) Find the value of x, y and z
(b) Calculate the composite index for these items for the year 2007 based on the year 2005
(c) The sales of these items for the year 2005 are RM 400.
Find the corresponding sales for the year 2007.
(d) Calculate the price index of item Q for the 2009 based on the year 2005, if its price increases at the same rate as it increases from the year 2005 to the year 2007.
6. Table 13 shows the prices and the price indices for four ingredients A, B, C and D used in making a type of biscuits. Diagram 13 shows a pie chart which represents the relative amount of the ingredients A, B, C and D used in making these biscuits.
| Ingredients | Price per kg (RM) | Price index for the year 2007 based on the year 2002 | |
| Year 2002 | Year 2007 | ||
| A | 1.80 | 2.25 | 125 |
| B | 3.00 | 4.20 | x |
| C | 0.80 | y | 150 |
| D | z | 0.60 | 80 |
Table 13
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Diagram 13
(a) Find the value of x, of y and of z
(b) (i) Calculate the composite index for the cost of making these biscuits in the year 2007 based on the year 2002
(ii)Hence, calculate the corresponding cost of making these biscuits in the year 2002 if the cost in the year 2007 was RM 3600
(c) The cost of making these biscuits is expected to increase by 10% from the year 2007 to the year 2009. Find the expected composite index for the year 2009 based on the year 2002
